
The convexification method.
The main idea of the convexification method to solve (REF ) is to minimize the Carleman weighted functional
\(U\mapsto \int _{\Omega } W_\lambda ({\bf x})|\Delta U - SU + \mathcal {F}({\bf x}, U, \nabla U)|^2d{\bf x}+ \mbox{a regularization term}\) 
subject to the given boundary conditions for some
Carleman weight function \(W_\lambda ({\bf x})\) .
With suitable choice of \(W_\lambda ({\bf x})\)  and the regularization term, one can prove that the functional in (REF ) is strictly convex in any bounded set of \(H^s(\Omega )^N\)  where \(s > {d/2} + 2.\)  The strict convexity implies that the minimizer is unique. Two other important theoretical results for the convexification method are (1) the minimizer can be obtained by using the conventional gradient descent method and the (2) the minimizer is an approximation of the desired solution to (REF ). The original idea about the convexification method is introduced in [1]}.
See [2]}, [3]}, [4]}, [5]} for follow-up results.
Although effective in delivering good numerical solutions, the convexification method has a drawback. It is time consuming.
We therefore do not employ the convexification method in this paper.

The Carleman contraction method.
The main idea of the contraction method to solve (REF ) is that we take an arbitrary function \(U_0 \in H^2(\Omega )^N\) .
Assume that \(U_n\)  is known, we compute \(U_{n + 1}\)  by solving
\(\left\lbrace \begin{array}{ll}\Delta U_{n+1} - SU_{n+1} + \mathcal {F}({\bf x}, U_n, \nabla U_{n}) = 0 & {\bf x}\in \Omega , \\U_{n+1}({\bf x}) = G_0({\bf x})& {\bf x}\in \partial \Omega ,\\\partial _{\nu } U_{n+1}({\bf x}) = G_1({\bf x}) &{\bf x}\in \partial \Omega .\end{array}\right.\) 
by using the quasi-reversibility method [6]} combining with a suitable Carleman weight function as in [7]}.
One can follow the arguments in [7]}, [9]} to prove the convergence of the constructed sequence \(\lbrace U_n\rbrace _{n \ge 0}\)  to the true solution to (REF ). For more details, see the following papers [10]}, [7]}, [9]}, [13]}.
This method was used to solve a similar inverse problem to Problem REF . We therefore do not repeat it in this paper.

 The Carleman method combining with linearization [14]}. We name this method Carleman-Newton method. Details of this method will be given in this section.

